垫江二中高中2018级数学竞赛试题 11页

重庆市垫江二中10-11学年高二上学期竞赛试题时间：150分钟满分：150分命题人：王超一.选择题：（本大题共10小题，每小题5分，共50分。在各题所给出的四个选项中，有且只有一个是正确的，请将正确选项的代号填在答题卡上）1.条件：“直线在y轴上的截距是在x轴上截距的两倍”；条件：“直线的斜率为－2”，则是的（）A．充分不必要条件B．必要不充分条件C．充要条件D．非充分也非必要条件2.已知椭圆的右焦点为,右准线为，点，线段交于点.若,则=()A.B.2C.D.3w.w.w.k.s.5.u.c.o3.函数的定义域为R，若与都是奇函数，则()w.w.w.k.s.5.u.c.o.mA.是偶函数B.是奇函数C.D.是奇函数4.已知，则有（）A最大值1.25B最小值1.25C最大值4D最小值15.已知A（-1，0）.B（1，0），点C（x，y）满足，则=（）A6B4C2D不能确定6.已知点的距离相等，则的最小值为（）A．2B．4C．D．7.在圆内过点（，）有条弦的长度成等差数列，最短弦长为数列首项,最长弦长为，若公差，那么的取值集合（）A．B．C．D.\n8.已知双曲线的准线过椭圆的焦点，则直线与椭圆至多有一个交点的充要条件是A.B.C.D.9.对于正实数，记为满足下述条件的函数构成的集合：且，有．下列结论中正确的是()A．若，，则B．若，，且，则C．若，，则w.w.w.k.s.5.u.c.o.mD．若，，且，则10.古希腊人常用小石子在沙滩上摆成各种形状来研究数。比如：w.w.w.k.s.5.u.c.o.m他们研究过图1中的1，3，6，10，…，由于这些数能够表示成三角形，将其称为三角形数；类似的，称图2中的1，4，9，16，…这样的数为正方形数。下列数中既是三角形数又是正方形数的是（）A.289B.1024C.1225D.1378二.填空题：（每小题5分，共20分，把答案填在题中横线上）11.对于椭圆和双曲线有下列命题：①椭圆的焦点恰好是双曲线的顶点;\n①双曲线的焦点恰好是椭圆的顶点;②双曲线与椭圆共焦点;③椭圆与双曲线有两个顶点相同.其中正确命题的序号是.12.如图，在平面直角坐标系中，为椭圆的四个顶点，为其右焦点，直线与直线相交于点T，线段与椭圆的交点恰为线段的中点，则该椭圆的离心率为.13.已知函数.项数为27的等差数列满足，且公差.若，则当=____________是，.14.将函数的图像绕坐标原点逆时针方向旋转角，得到曲线.若对于每一个旋转角，曲线都是一个函数的图像，则的最大值为__________15.已知数列满足：（m为正整数），若，则m所有可能的取值为__________.三.解答题：（共6题，总分75分）16.（本题满分12分）垫江县某商场为使销售空调和冰箱获得的总利润达到最大，对即将出售的空调和冰箱相关数据进行调查，得出下表：资金每台空调或冰箱所需资金（百元）月资金供应数量（百元）空调冰箱成本3020300工人工资510110每台利润68问：该商场怎样确定空调或冰箱的月供应量，才能使总利润最大？17.（本题满分12分）某地街道呈现东—西.南—北向的网格状，相邻街距都为1.两街道相交的点称为格点。若以互相垂直的两条街道为轴建立直角坐标系，现有下述格点，，，，，为报刊零售点.请确定一个格点（除零售点外）为发行站，使6个零售点沿街道到发行站之间路程的和最短.并求出最短路程.18.（本题满分12分）按照某学者的理论，假设一个人生产某产品单件成本为\n元，如果他卖出该产品的单价为元，则他的满意度为；如果他买进该产品的单价为元，则他的满意度为.如果一个人对两种交易(卖出或买进)的满意度分别为和，则他对这两种交易的综合满意度为.现假设甲生产A、B两种产品的单件成本分别为12元和5元，乙生产A、B两种产品的单件成本分别为3元和20元，设产品A、B的单价分别为元和元，甲买进A与卖出B的综合满意度为，乙卖出A与买进B的综合满意度为(1)求和关于、的表达式；当时，求证：=；(2)设，当、分别为多少时，甲、乙两人的综合满意度均最大？最大的综合满意度为多少？(3)记(2)中最大的综合满意度为，试问能否适当选取、的值，使得和同时成立，但等号不同时成立？试说明理由。19．(本题满分12分)设为实数，函数.(1)若，求的取值范围；(2)求的最小值；(3)设函数，直接写出(不需给出演算步骤)不等式的解集.20.（本题满分13分）设椭圆E:（a,b>0）过M（2，），N(,1)两点，O为坐标原点，（I）求椭圆E的方程；（II）是否存在圆心在原点的圆，使得该圆的任意一条切线与椭圆E恒有两个交点A,B,且？若存在，写出该圆的方程，并求|AB|的取值范围，若不存在说明理由。21.（本题满分14分）已知是公差为的等差数列，是公比为的等比数列.(1)若，是否存在，有说明理由；(2)找出所有数列和，使对一切,,并说明理由；\n(1)若试确定所有的，使数列中存在某个连续项的和是数列中的一项，请证明。重庆市垫江二中10-11学年高二上学期竞赛试题参考答案一.选择题1-5BADDB6-10BCACC二.填空题11.①②12.13.1414.15.4532三.解答题16.设空调和冰箱的月供应量分别为台，月总利润为百元则………………………………………3分作出可行域……………………………………………………………………………6分，纵截距为，斜率为k=，满足欲最大，必最大，此时，直线必过图形的一个交点（4，9），分别为4，9∴空调和冰箱的月供应量分别为4、9台时，月总利润为最大.……………12分17.解：设发行站的位置为，零售点到发行站的距离为这六个点的横纵坐标的平均值为，，记A（2，）\n画出图形可知，发行站的位置应该在点A附近，代入附近的点的坐标进行比较可知，在（3,3）处z取得最小值.即.18.解：（1）当时，,,=（2）当时，由，故当即时，甲乙两人同时取到最大的综合满意度为。（3）由（2）知：=由得：，令则，即：。同理，由得：另一方面，当且仅当，即=时，取等号。\n所以不能否适当选取、的值，使得和同时成立，但等号不同时成立。19.解：（1）若，则（2）当时，当时，综上（3）时，得，当时，；当时，△>0,得：讨论得：当时，解集为;当时，解集为;当时，解集为.20.解:（1）因为椭圆E:（a,b>0）过M（2，），N(,1)两点,所以解得所以椭圆E的方程为（2）假设存在圆心在原点的圆，使得该圆的任意一条切线与椭圆E恒有两个交点A,B,且,设该圆的切线方程为解方程组得\n,即,则△=,即要使,需使,即,所以,所以又,所以,所以,即或,因为直线为圆心在原点的圆的一条切线,所以圆的半径为,,,所求的圆为,此时圆的切线都满足或,而当切线的斜率不存在时切线为与椭圆的两个交点为或满足,综上,存在圆心在原点的圆，使得该圆的任意一条切线与椭圆E恒有两个交点A,B,且.因为,\n所以,,①当时因为所以,所以,所以当且仅当时取”=”.②当时,.③当AB的斜率不存在时,两个交点为或,所以此时,综上,|AB|的取值范围为即:21.[解法一]（1）由，得，．．．．．．2分整理后，可得，.，为整数，不存在.，使等式成立。．．．．．．4分（2）若，即，（*）（ⅰ）若则。\n当{}为非零常数列，{}为恒等于1的常数列，满足要求。．．．．．．6分（ⅱ）若，（*）式等号左边取极限得，（*）式等号右边的极限只有当时，才能等于1。此时等号左边是常数，，矛盾。综上所述，只有当{}为非零常数列，{}为恒等于1的常数列，满足要求。．．．．．．8分【解法二】设则（i）若d=0，则（ii）若（常数）即，则d=0，矛盾综上所述，有，．．．．．．8分（3）设.，.．．．．．．10分取．．．．．．12分由二项展开式可得正整数M1.M2，使得（4-1）2s+2=4M1+1,故当且仅当p=3s,sN时，命题成立.．．．．．．14分\n内部资料仅供参考9JWKffwvG#tYM*Jg&6a*CZ7H$dq8KqqfHVZFedswSyXTy#&QA9wkxFyeQ^!djs#XuyUP2kNXpRWXmA&UE9aQ@Gn8xp$R#͑Gx^Gjqv^$UE9wEwZ#Qc@UE%&qYp@Eh5pDx2zVkum&gTXRm6X4NGpP$vSTT#&ksv*3tnGK8!z89AmYWpazadNu##KN&MuWFA5uxY7JnD6YWRrWwc^vR9CpbK!zn%Mz849Gx^Gjqv^$UE9wEwZ#Qc@UE%&qYp@Eh5pDx2zVkum&gTXRm6X4NGpP$vSTT#&ksv*3tnGK8!z89AmYWpazadNu##KN&MuWFA5ux^Gjqv^$UE9wEwZ#Qc@UE%&qYp@Eh5pDx2zVkum&gTXRm6X4NGpP$vSTT#&ksv*3tnGK8!z89AmYWpazadNu##KN&MuWFA5uxY7JnD6YWRrWwc^vR9CpbK!zn%Mz849Gx^Gjqv^$UE9wEwZ#Qc@UE%&qYp@Eh5pDx2zVkum&gTXRm6X4NGpP$vSTT#&ksv*3tnGK8!z89AmUE9aQ@Gn8xp$R#͑Gx^Gjqv^$UE9wEwZ#Qc@UE%&qYp@Eh5pDx2zVkum&gTXRm6X4NGpP$vSTT#&ksv*3tnGK8!z89AmYWpazadNu##KN&MuWFA5uxY7JnD6YWRrWwc^vR9CpbK!zn%Mz849Gx^Gjqv^$UE9wEwZ#Qc@UE%&qYp@Eh5pDx2zVkum&gTXRm6X4NGpP$vSTT#&ksv*3tnGK8!z89AmYWpazadNu##KN&MuWFA5ux^Gjqv^$UE9wEwZ#Qc@UE%&qYp@Eh5pDx2zVkum&gTXRm6X4NGpP$vSTT#&ksv*3tnGK8!z89AmYWpazadNu##KN&MuWFA5uxY7JnD6YWRrWwc^vR9CpbK!zn%Mz849Gx^Gjqv^$UE9wEwZ#Qc@UE%&qYp@Eh5pDx2zVkum&gTXRm6X4NGpP$vSTT#&ksv*3tnGK8!z8vG#tYM*Jg&6a*CZ7H$dq8KqqfHVZFedswSyXTy#&QA9wkxFyeQ^!djs#XuyUP2kNXpRWXmA&UE9aQ@Gn8xp$R#͑Gx^Gjqv^$UE9wEwZ#Qc@UE%&qYp@Eh5pDx2zVkum&gTXRm6X4NGpP$vSTT#&ksv*3tnGK8!z89AmYWpazadNu##KN&MuWFA5uxY7JnD6YWRrWwc^vR9CpbK!zn%Mz849Gx^G89AmUE9aQ@Gn8xp$R#͑Gx^Gjqv^$UE9wEwZ#Qc@UE%&qYp@Eh5pDx2zVkum&gTXRm6X4NGpP$vSTT#&ksv*3tnGK8!z89AmYWpazadNu##KN&MuWFA5uxY7JnD6YWRrWwc^vR9CpbK!zn%Mz849Gx^Gjqv^$UE9wEwZ#Qc@UE%&qYp@Eh5pDx2zVkum&gTXRm6X4NGpP$vSTT#&ksv*3tnGK8!z89AmYWpazadNu##KN&MuWFA5ux^Gjqv^$UE9wEwZ#Qc@UE%&qYp@Eh5pDx2zVkum&gTXRm6X4NGpP$vSTT#&ksv*3tnGK8!z89AmYWpazadNu##KN&MuWFA5uxY7JnD6YWRrWwc^vR9CpbK!zn%Mz849Gx^Gjqv^$UE9wEwZ#Qc@UE%&qYp@Eh5pDx2zVkum&gTXRm6X4NGpP$vSTT#&ksv*3tnGK8!z8vG#tYM*Jg&6a*CZ7H$dq8KqqfHVZFedswSyXTy#&QA9wkxFyeQ^!djs#XuyUP2kNXpRWXmA&UE9aQ@Gn8xp$R#͑Gx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